unifed.kappa {unifed} | R Documentation |
Cumulant generator of the unifed distribution
Description
Cumulant generator of the unifed distribution
Usage
unifed.kappa(theta)
unifed.kappa.prime(theta)
unifed.kappa.double.prime(theta)
unifed.kappa.prime.inverse(mu, ...)
unifed.kappa.prime.inverse.one(mu, tol = 1e-07, maxit = 1e+07)
Arguments
theta |
A numeric vector. |
mu |
A vector of numbers between 0 and 1 |
... |
Other parameters of |
tol |
Tolerance level. The algorithm stops if the proportional difference between the new and old value of an iteration is less or equal than this number. |
maxit |
Maximum number of iterations of the algorithm to look for convergence. |
Details
The cumulant generator of the unifed distribution is defined as
\kappa(\theta)=\left\{
\begin{array}{ll}
\log\left(\frac{e^\theta-1}{\theta}\right)& if \theta \neq
0\\
0 & \mbox{if }\theta=0
\end{array}
\right..
unifed.kappa.prime.inverse.one
uses the
Newthon-Raphson method for finding the inverse of
unifed.kappa.prime
for a single value.
Value
unifed.kappa
returns a vector that contains the
cumulant generator of the unifed distribution applied to each
element of theta.
unifed.kappa.prime
returns a vector that contains
the derivative of the cumulant generator of the unifed
distribution for each element of theta.
unifed.kappa.double.prime
returns a vector that
contains the second derivative of the cumulant generator of the
unifed distribution for each element of theta.
unifed.kappa.prime.inverse
returns a vector with
unifed.kappa.prime.inverse.one
evaluated at every entry
of mu
.
unifed.kappa.prime.inverse.one
if the tolerance
level is reached within maxit
iterations, the function
returns the value of the last iteration. Otherwise it returns
NA
.
References
Quijano Xacur, O.A. The unifed distribution. J Stat Distrib App 6, 13 (2019). doi:10.1186/s40488-019-0102-6.
Jørgensen, Bent (1997). The Theory of Dispersion Models. Chapman & Hall, London.
Examples
unifed.kappa(1)
unifed.kappa(-5:5)
unifed.kappa.prime(4.5)
unifed.kappa.double.prime(0)
unifed.kappa.prime.inverse(0.5)
unifed.kappa.prime.inverse(c(0.1,0.7,0.9))